Heat Kernels on Metric Measure Spaces

Heat kernels on metric measure spaces

A.Grigor’yan

Lectures at Cornell Probability Summer School, July 2010 2 Contents

1 The notion of the heat kernel 5 1.1 Examples of heat kernels ...... 5 1.1.1 Heat kernel in Rn ...... 5 1.1.2 Heat kernels on Riemannian manifolds ...... 6 1.1.3 Heat kernels of fractional powers of Laplacian ...... 7 1.1.4 Heat kernels on fractal spaces ...... 8 1.1.5 Summary of examples ...... 9 1.2 Abstract heat kernels ...... 10 1.3 A heat semigroup ...... 11 1.4 The Dirichlet form ...... 12 1.5 More examples of heat kernels ...... 15 1.6 Summary of Chapter 1 ...... 17

2 Consequences of heat kernel bounds 19 2.1 Identifying Φ in the non-local case ...... 19 2.2 Volume of balls ...... 21 2.3 Besov spaces ...... 23 2.4 Subordinated semigroup ...... 24 2.5 The walk dimension ...... 25 2.6 Inequalities for the walk dimension ...... 27 2.7 Identifying Φ in the local case ...... 32

3 Upper bounds of the heat kernel 35 3.1 Ultracontractive semigroups ...... 35 3.2 Restriction of the Dirichlet form ...... 36 3.3 Faber-Krahn and Nash inequalities ...... 37 3.4 Oﬀ-diagonal upper bounds ...... 40

4 Two-sided bounds of the heat kernel 47 4.1 Using elliptic Harnack inequality ...... 47 4.2 Matching upper and lower bounds ...... 49 4.3 Further results ...... 53

3 4 CONTENTS Chapter 1

The notion of the heat kernel

1.1 Examples of heat kernels

1.1.1 Heat kernel in Rn The classical Gauss-Weierstrass heat kernel is the following function

1 x y 2 p x, y , t ( ) = n/2 exp | − | ((1)) (4πt) − 4t ! where x, y Rn and t > 0. This function is a fundamental solution of the heat equation ∈ ∂u = ∆u, ∂t n ∂2 where ∆ = i 2 is the Laplace operator. Moreover, if f is a continuous bounded =1 ∂xi function on n then the function PR

u (t, x) = pt (x, y) f (y) dy n ZR solves the Cauchy problem ∂u = ∆u ∂t . u (0, x) = f (x) This also can be written in the form

u (t, ) = exp ( t ) f , · − L where here is a self-adjoint extension of ∆ in L2 (Rn) and exp ( t ) is understood in theL sense of the functional calculus of− self-adjoint operators.− ThisL means that pt (x, y) is the integral kernel of the operator exp ( t ). − L The function pt (x, y) has also a probabilistic meaning: it is the transition density n of Brownian motion Xt t 0 in R . The graph of pt (x, 0) as a function of x is shown here: { } ≥

5 6 CHAPTER 1. THE NOTION OF THE HEAT KERNEL

0.75

0.5

0.25

0 -5 -2.5 0 2.5 5

Figure 1.1: The Gauss-Weierstrass heat kernel at diﬀerent values of t

x y 2 2 The term | − | determines the space/time scaling: if x y Ct then pt (x, y) t | − | ≤ is comparable with pt (x, x), that is, the probability density in the C√t-neighborhood of x is nearly constant.

1.1.2 Heat kernels on Riemannian manifolds Let (M, g) be a connected Riemannian manifold, and ∆ be the Laplace-Beltrami operator on M. Then the heat kernel pt (x, y) is can be deﬁned as the integral kernel of the heat semigroup exp ( t ) t 0 where is the Dirichlet Laplace operator, that is, the minimal self-adjoint{ extension− L } ≥ of ∆L in L2 (M, µ), and µ is the Riemannian − volume. Alternatively, pt (x, y) is the minimal positive fundamental solution of the heat equation ∂u = ∆u. ∂t

The function pt (x, y) can be used to deﬁne Brownian motion Xt t 0 on M. Namely, { } ≥ Xt t 0 is a diﬀusion process (that is, a Markov process with continuous trajecto- {ries),} ≥ such that

Px (Xt A) = pt (x, y) dµ (y) ∈ ZA for any Borel set A M. Let d (x, y) be the⊂ geodesic distance on M. It turns out that the Gaussian type d2(x,y) space/time scaling t appears in heat kernel estimates on general Riemannian manifolds: 1. (Varadhan) For an arbitrary Riemannian manifold, d2 (x, y) log pt (x, y) as t 0, ∼ − 4t → 2. (Davies) For an arbitrary manifold M, for any two measurable sets A, B M ⊂ d2 (A, B) pt (x, y) dµ (x) dµ (y) µ (A) µ (B) exp ≤ − 4t A B Z Z p 1.1. EXAMPLES OF HEAT KERNELS 7

x X t A

Figure 1.2: The Brownian motion Xt hits a set A

Technically, all these results depend upon the following property of the geodesic distance: d 1. It is natural|∇ | ≤ to ask the following question:

Are there settings where the space/time scaling is diﬀerent from Gaussian?

1.1.3 Heat kernels of fractional powers of Laplacian Easy examples can be constructed using another operator instead of the Laplacian. As above, let be the Dirichlet Laplace operator on a Riemannian manifold M, and consider theL evolution equation ∂u + β/2u = 0, ∂t L where β (0, 2). The operator β/2 is understood in the sense of the functional ∈ 2 L β/2 calculus in L (M, µ) . Let pt (x, y) be now the heat kernel of , that is, the integral kernel of exp t β/2 . L − L The condition β < 2 leads to the fact that the semigroup exp t β/2 is Markovian, which, in particular, means that p (x, y) > 0 (if β > 2 then− Lp (x, y) t t may be signed). Using the techniques of subordinators, one obtains the following estimate for the heat kernel of β/2 in Rn: L n+β (n+β) β β C x y − C x y − pt (x, y) 1 + | − | 1 + | − | . ((2)) tn/β t1/β tn/β t ! (the symbol means that both and are valid but with diﬀerent values of the constant C). ≤ ≥ The heat kernel of √ in Rn (that is, the case β = 1) is known explicitly: L n+1 2 c x y − 2 c t p x, y n n , t( ) = n 1 + | −2 | = n+1 t t 2 2 t2 + x y | − | 8 CHAPTER 1. THE NOTION OF THE HEAT KERNEL

n+1 (n+1)/2 where cn = Γ 2 /π . This function coincides with the Poisson kernel in n+1 n the half-space R+ and with the density of the Cauchy distribution in R with the parameter t. dβ (x,y) As we see, the space/time scaling is given by the term t where β < 2. The heat kernel of the operator β/2 is the transition density of a symmetric stable process of index β that belongsL to the family of Levy processes. The trajectories of this process are discontinuous, thus allowing jumps. The heat kernel pt (x, y) of such process is nearly constant in a Ct1/β-neighborhood of y. If t is large then

t1/β t1/2, that is, this neighborhood is much larger than that for the diﬀusion process, which is not surprising because of the presence of jumps. The space/time scaling with β < 2 is called super-Gaussian.

1.1.4 Heat kernels on fractal spaces A rich family of heat kernels for diﬀusion processes has come from Analysis on fractals. Loosely speaking fractals are subsets of Rn with certain self-similarity properties. One of the best understood fractals is the Sierpinski gasket (SG). The construction of the Sierpinski gasket is similar to the Cantor set: one starts with a triangle as a closed subset of R2, then eliminates the open middle triangle (shaded on the diagram), then repeats this procedure for the remaining triangles, etc.

Figure 1.3: Construction of the Sierpinski gasket

Hence, SG is a compact connected subset of R2. The unbounded SG is obtained from SG by merging the latter (at the left lower corner of the next diagram) with two shifted copies and then by repeating this procedure at larger scales. Barlow and Perkins ’88, Goldstein ’87 and Kusuoka ’87 have independently constructed by different methods a natural diffusion process on SG that has the same self-similarity as SG. Barlow and Perkins considered random walks on the graph approximations of SG and showed that, with an appropriate scaling, the random walks converge to a diffusion process. Moreover, they proved that this process has 1.1. EXAMPLES OF HEAT KERNELS 9

Figure 1.4: The unbounded SG is obtained from SG by merging the latter (at the left lower corner of the diagram) with two shifted copies and then by repeating this procedure at larger scales.

a transition density pt (x, y) with respect to a proper Hausdorﬀ measure µ of SG, and that pt satisﬁes the following estimate:

1 β β 1 C d (x, y) − pt (x, y) exp c , ((3)) tα/β − t ! where d (x, y) = x y and | − | log 3 log 5 α = dim SG = , β = > 2. H log 2 log 2

Similar estimates were proved by Barlow and Bass for other families of fractals, including Sierpinski carpets, and the parameters α and β in (3) are determined by the intrinsic properties of the fractal. In all cases, α is the Hausdorﬀ dimension (which is also called the fractal dimension). The parameter β, that is called the walk dimension, is larger than 2 in all interesting examples. 1/β The heat kernel pt (x, y), satisfying (3) is nearly constant in a Ct -neighborhood of y. If t is large then t1/β t1/2, that is, this neighborhood is much smaller than that for the diﬀusion process, which is due to the presence of numerous holes-obstacles that the Brownian particle must bypass. The space/time scaling with β > 2 is called sub-Gaussian.

1.1.5 Summary of examples Observe now that in all the above examples, the heat kernel estimates can be uniﬁed as follows: 10 CHAPTER 1. THE NOTION OF THE HEAT KERNEL

C d (x, y) pt (x, y) Φ c , ((4)) tα/β t1/β where α, β are positive parameters and Φ (s) is a positive decreasing function on [0, + ). For example, the Gauss-Weierstrass function (1) satisfies (4) with α = n, β = 2∞ and Φ(s) = exp s2 − (Gaussian estimate). The heat kernel (2) of the symmetric stable process in Rn satisfies (4) with α = n, 0 < β < 2, and (α+β) Φ(s) = (1 + s)− (super-Gaussian estimate). The heat kernel (3) of diffusions on fractals satisfies (4) with β > 2 and

β Φ(s) = exp s β 1 − − (sub-Gaussian estimate). There are at least two questions related to the estimates of the type (4):

1. What values of the parameters α, β and what functions Φ can actually occur in the estimate (4)?

2. How to obtain estimates of the type (4)?

To give these question a precise meaning, we must deﬁne what is a heat kernel.

1.2 Abstract heat kernels

Let (M, d) be a locally compact separable metric space and µ be a Radon measure on M with full support. The triple (M, d, µ) will be called a metric measure space.

Deﬁnition. A family pt t>0 of measurable functions pt(x, y) on M M is called a heat kernel if the following{ } conditions are satisﬁed, for µ-almost all ×x, y M and all s, t > 0: ∈

(i) Positivity: pt (x, y) 0. ≥ (ii) The total mass inequality:

pt(x, y)dµ(y) 1. ≤ ZM

(iii) Symmetry: pt(x, y) = pt(y, x). 1.3. A HEAT SEMIGROUP 11

(iv) The semigroup property:

ps+t(x, y) = ps(x, z)pt(z, y)dµ(z). ZM (v) Approximation of identity: for any f L2 := L2 (M, µ), ∈ L2 pt(x, y)f(y)dµ(y) f(x) as t 0 + . −→ → ZM

If in addition we have, for all t > 0 and almost all x M, ∈

pt(x, y)dµ(y) = 1 ZM then the heat kernel pt is called stochastically complete (or conservative).

1.3 A heat semigroup

Any heat kernel gives rise to the family of operators Pt t 0 where P0 = id and Pt for t > 0 is deﬁned by { } ≥

Ptf(x) = pt(x, y)f(y)dµ(y), ZM where f is a measurable function on M. It follows from (i) (ii) that the operator − Pt is Markovian, that is, f 0 implies Ptf 0 and f 1 implies Ptf 1. It ≥ 2 ≥ ≤ ≤ follows that Pt is a bounded operator in L and, moreover, is a contraction, that is,

Pt L2 L2 1. k k → ≤ The symmetry property (iii) implies that the operator Pt is symmetric and, hence, self-adjoint. The semigroup property (iv) implies that PtPs = Pt+s, that is, the family Pt t 0 is a semigroup of operators. It follows from (v) that { } ≥

s- lim Pt = id = P0 t 0 → where s-lim stands for the strong limit. Hence, Pt t 0 is a strongly continuous, 2 { } ≥ symmetric, Markovian semigroup in L . We say shortly that Pt is a heat semigroup. { } Conversely, if Pt is a heat semigroup and if it has the integral kernel pt (x, y) then the latter is{ a heat} kernel in the sense of the above Definition. 2 Given a heat semigroup Pt in L , define the infinitesimal generator of the semigroup by L f Ptf f := lim − , t 0 t L → 12 CHAPTER 1. THE NOTION OF THE HEAT KERNEL where the limit is understood in the L2-norm. The domain dom( ) of the generator is the space of functions f L2 for which the above limit exists.L By the Hille– YosidaL theorem, dom( ) is dense∈ in L2. Furthermore, is a self-adjoint, positive L L definite operator, which immediately follows from the fact that the semigroup Pt { } is self-adjoint and contractive. Moreover, Pt can be recovered from as follows L

Pt = exp ( t ) , − L where the right hand side is understood in the sense of spectral theory. Heat kernels and heat semigroups arise naturally from Markov processes. Let

Xt t 0 , Px x M be a Markov process on M, that is reversible with respect to measure{ } ≥ µ{. Assume} ∈ that it has the transition density p (x, y), that is, a function t such that, for all x M, t > 0, and all Borel sets A M, ∈ ⊂

Px (Xt A) = pt (x, y) dµ (y) . ∈ ZM Then pt (x, y) is a heat kernel in the sense of the above Deﬁnition.

1.4 The Dirichlet form

Given a heat semigroup Pt on a metric measure space (M, d, µ), define for any { }2 t > 0 a bilinear form t on L by E u Ptu 1 t (u, v) := − , v = ((u, v) (Ptu, v)) , E t t − 2 where ( , ) is the inner product in L . Since Pt is symmetric, the form t is also · · E symmetric. Since Pt is a contraction, it follows that 1 t (u) := t (u, u) = ((u, u) (Ptu, u)) 0, E E t − ≥ that is, t is a positive definite form. E In terms of the spectral resolution Eλ of the generator , t can be expressed as follows { } L E tλ 1 1 ∞ 2 ∞ tλ 2 ∞ 1 e− 2 t (u) = ((u, u) (Ptu, u)) = d Eλu e− d Eλu = − d Eλu , E t − t k k2 − k k2 t k k2 Z0 Z0 Z0 tλ 1 e− which implies that t (u) is decreasing in t, since the function t − t is decreasing. Define for any Eu L2 7→ ∈ (u) = lim t (u) t 0 E ↓ E where the limit (finite or infinite) exists by the monotonicity, so that (u) t (u). 1 e tλ E ≥ E Since − − λ as t 0, we have t → → ∞ 2 (u) = λd Eλu . E k k2 Z0 1.4. THE DIRICHLET FORM 13

Set : = u L2 : (u) < = dom 1/2 dom ( ) F { ∈ E ∞} L ⊃ L and deﬁne a bilinear form (u, v) on by the polarization identity E F 1 (u, v) := ( (u + v) (u v)) , E 4 E − E − which is equivalent to (u, v) = lim t (u, v) . t 0 E → E Note that contains dom( ). Indeed, if u dom( ) then we have for all v L2 F L ∈ L ∈ u Ptu lim t (u, v) = lim − , v = ( u, v) . t 0 E t 0 t L → → Setting v = u we obtain u . Then choosing any v we obtain the identity ∈ F ∈ F (u, v) = ( u, v) for all u dom( ) and v . E L ∈ L ∈ F The space is naturally endowed with the inner product F [u, v] := (u, v) + (u, v) . E It is possible to show that the form is closed, that is, the space is Hilbert. Furthermore, dom ( ) is dense in . E F L F The fact that Pt is Markovian implies that the form is also Markovian, that is E

u u := min(u+, 1) and (u) (u) . ∈ F ⇒ ∈ F E ≤ E Indeed, let us ﬁrst show that for any u L2 e ∈ e

t (u+) t (u) . E ≤ E We have

t (u) = t (u+ u ) = t (u+) + t (u ) 2 t (u+, u ) t (u+) E E − − E E − − E − ≥ E because t (u ) 0 and E − ≥ 1 1 t (u+, u ) = (u+, u ) (Ptu+, u ) 0. E − t − − t − ≤ Assuming u and letting t 0 we obtain ∈ F →

(u+) = lim t (u+) lim t (u) = (u) < t 0 t 0 E → E ≤ → E E ∞ whence (u+) (u) and, hence, u+ . E ≤ E ∈ F Similarly one proves that u = min(u+, 1) belongs to and (u) (u+). F E ≤ E e e 14 CHAPTER 1. THE NOTION OF THE HEAT KERNEL

Conclusion. Hence, ( , ) is a Dirichlet form, that is, a bilinear, symmetric, positive deﬁnite, closed, denselyE F deﬁned form in L2 with Markovian property.

If the heat semigroup is deﬁned by means of a heat kernel pt, then t can be equivalently deﬁned by E

1 2 1 2 t (u) = (u(x) u(y)) pt(x, y)dµ(y)dµ(x)+ (1 Pt1(x)) u (x)dµ(x). E 2t M M − t M − Z Z Z ((5)) Indeed, we have

u(x) Ptu(x) = u (x) Pt1 (x) Ptu (x) + (1 Pt1(x)) u (x) − − − = (u(x) u(y)) pt(x, y)dµ(y) + (1 Pt1(x)) u (x) − − ZM whence 1 t (u) = (u(x) u(y)) u(x)pt(x, y)dµ(y)dµ(x) E t − ZM ZM 1 2 + (1 Pt1(x)) u (x)dµ(x). t − ZM Interchanging the variables x and y in the ﬁrst integral and using the symmetry of the heat kernel, we obtain also 1 t (u) = (u(y) u(x)) u(y)pt(x, y)dµ(y)dµ(x) E t − ZM ZM 1 2 + (1 Pt1(x)) u (x)dµ(x), t − ZM and (5) follows by adding up the two previous lines. Note that Pt1 1 so that the second term in the right hand side of (5) is non- ≤ negative. If the heat kernel is stochastically complete, that is, Pt1 = 1, then that term vanishes and we obtain

1 2 t (u) = (u(x) u(y)) pt(x, y)dµ(y)dµ(x). ((6)) E 2t − ZM ZM Deﬁnition. The form ( , ) is called local if (u, v) = 0 whenever the functions u, v have compact disjointE F supports. The formE ( , ) is called strongly local if (u,∈ v) F = 0 whenever the functions u, v have compactE F supports and u const inE an open neighborhood of supp v. ∈ F ≡

For example, if pt (x, y) is the heat kernel of the Laplace-Beltrami operator on a complete Riemannian manifold then the associated Dirichlet form is given by

(u, v) = u, v dµ, ((7)) E h∇ ∇ i ZM 1.5. MORE EXAMPLES OF HEAT KERNELS 15

1 and is the Sobolev space W2 (M). Note that this Dirichlet form is strongly local becauseF u = const on supp v implies u = 0 on supp v and, hence, (u, v) = 0. ∇ E n If pt (x, y) is the heat kernel of the symmetric stable process of index β in R , that is, = ( ∆)β/2, then L − (u (x) u (y)) (v (x) v (y)) (u, v) = cn,β − n+β − dxdy, E n n x y ZR ZR | − | β/2 n 2 and is the Besov space B2,2 (R ) = u L : (u, u) < . This form is clearly non-local.F { ∈ E ∞} Denote by C0 (M) the space of continuous functions on M with compact supports, endowed with sup-norm.

Definition. The form ( , ) is called regular if C0 (M) is dense both in E F F ∩ F (with [ , ]-norm) and in C0 (M) (with sup-norm). · · All the Dirichlet forms in the above examples are regular. Assume that we are given a Dirichlet form ( , ) in L2 (M, µ). Then one can define the generator of ( , ) by the identity E F L E F ( u, v) = (u, v) for all u dom ( ) , v ((8)) L E ∈ L ∈ F where dom ( ) must satisfy one of the following two equivalent requirements: L ⊂ F 1. dom ( ) is a maximal possible subspace of such that (8) holds L F 2. is a densely defined self-adjoint operator. L Clearly, is positive definite so that spec [0, + ). Hence, the family of L t L ⊂ ∞ operators Pt = e− L, t 0, forms a strongly continuous, symmetric, contraction semigroup in L2. Moreover,≥ using the Markovian property of the Dirichlet form ( , ), it is possible to prove that Pt is Markovian, that is, Pt is a heat semi- E F { } { } group. The question whether and when Pt has the heat kernel requires additional investigation.

1.5 More examples of heat kernels

Let us give some examples of stochastically complete heat kernels that do not satisfy (4).

Example. (A frozen heat kernel) Let M be a countable set and let xk ∞ be the { }k=1 sequence of all distinct points from M. Let µ ∞ be a sequence of positive reals { k}k=1 and deﬁne measure µ on M by µ ( xk ) = µk. Deﬁne a function pt (x, y) on M M by { } × 1 , x = y = xk p (x, y) = µk t 0, otherwise. 16 CHAPTER 1. THE NOTION OF THE HEAT KERNEL

Easy to check that pt (x, y) is a heat kernel. For example, let us check the approximation of identity: for any function f L2 (M, µ), we have ∈

Ptf (x) = pt (x, y) f (y) dµ (y) = pt (x, x) f (x) µ ( x ) = f (x) . { } ZM This identity implies also the stochastic completeness. The Dirichlet form is

f Ptf (f) = lim − , f = 0. E t 0 t → The Markov process associated with the frozen heat kernel is very simple: Xt = X0 for all t 0 so that it is a frozen diﬀusion. ≥ Example. (The heat kernel in H3) The heat kernel of the Laplace-Beltrami operator on the 3-dimensional hyperbolic space H3 is given by the formula 1 r r2 pt(x, y) = exp t , 3/2 sinh r −4t − (4πt) where r = d (x, y) is the geodesic distance between x, y. The Dirichlet form is (7). Example. (The Mehler heat kernel) Let M = R, measure µ be deﬁned by

2 dµ = ex dx, and the operator be given by L 2 x2 d x2 d d d = e− e = 2x . L − dx dx −dx2 − dx Then the heat kernel of is given by the formula L 1 2xye 2t x2 y2 p x, y − t . t ( ) = 1/2 exp − 4t − (2π sinh 2t) 1 e− − − The associated Dirichlet form is given by (7) . Similarly, for the measure x2 dµ = e− dx and for the operator

2 x2 d x2 d d d = e e− = + 2x , L dx dx −dx2 dx we have 1 2xye 2t (x2 + y2) e 4t p x, y − − t . t ( ) = 1/2 exp − 4t + (2π sinh 2t) 1 e− − 1.6. SUMMARY OF CHAPTER 1 17 1.6 Summary of Chapter 1

Given a metric measure space (M, d, µ), we have deﬁned the notion of a heat kernel pt (x, y) as a family of functions on M M that satisﬁes certain properties. It gives 2 × 2 rise to the family of operators Pt : L L →

Ptf = pt (x, y) f (y) dµ (y) ZM that forms a heat semigroup Pt t 0. The latter determines the generator , defined by { } ≥ L u Ptu u = L2- lim − t 0 t L → with dom ( ) = u L2: the above limit exists , which is a positive definite self- adjoint operatorL in{ L∈2. } The heat semigroup determines also a Dirichlet form ( , ) where as a quadratic form is defined by E F E

u Ptu (u) := lim t (u) = lim − , u E t 0 E t 0 t → → and = u L2:the above limit is ﬁnite dom ( ). F { ∈ } ⊃ L Note that t (u) monotone increases as t . In terms of the heat kernel we have E ↓

1 2 1 2 t (u) = (u(x) u(y)) pt(x, y)dµ(y)dµ(x)+ (1 Pt1(x)) u (x)dµ(x). E 2t − t − ZM ZM ZM Conversely, given a Dirichlet form ( , ) in L2, one defines the generator by the identity E F L ( u, v) = (u, v) for all u dom ( ) , v L E ∈ L ∈ F and the requirement that dom ( ) must be a maximal possible subspace of . Then L t F the heat semigroup is defined by Pt = e− L. The existence of the heat kernel (=the integral kernel of Pt) requires additional investigation. The Dirichlet form (or the heat semigroup) determines a Markov process ( Xt , Px ) on M such that { } { } Ex (f (Xt)) = Ptf (x) for all f b (M), t > 0 and almost all x M. The process Xt is a diffusion if and only if ( ∈, B) is local. ∈ We haveE F seen examples of heat kernels satisfying the estimates of the type

C d (x, y) pt (x, y) Φ c tα/β t1/β with the following functions Φ: 18 CHAPTER 1. THE NOTION OF THE HEAT KERNEL

1. Φ (s) = exp ( s2) (Gaussian estimates, Brownian motion in Rn) − (α+β) 2. Φ (s) = (1 + s)− , 0 < β < 2 (super-Gaussian estimates, symmetric stable processes in Rn)

β 3. Φ (s) = exp( s β 1 ), β > 2 (sub-Gaussian estimates, diﬀusions on fractals) − − Chapter 2

Consequences of heat kernel bounds

In this Chapter we assume that pt (x, y) is a heat kernel on a metric measure space (M, d, µ) that satisﬁes certain upper and/or lower estimates, and investigate the consequences of these estimates.

2.1 Identifying Φ in the non-local case

Fix two positive parameters α and β and a monotone decreasing function Φ : [0, + ) [0, + ). ∞ → ∞ Theorem 2.1. (AG, T.Kumagai ’09) Let pt be a heat kernel on (M, d, µ).

(a) If the heat kernel satisﬁes the estimate

1 d (x, y) pt (x, y) Φ , ≤ tα/β t1/β for all t > 0 and almost all x, y M, then either the associated Dirichlet form ( , ) is local or ∈ E F (α+β) Φ(s) c (1 + s)− ≥ for all s > 0 and some c > 0.

(b) If the heat kernel satisﬁes the estimate

1 d (x, y) pt (x, y) Φ , ≥ tα/β t1/β then (α+β) Φ(s) C (1 + s)− ≤ for all s > 0 and some C > 0.

19 20 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS

(c) Consequently, if the heat kernel satisﬁes the estimate C d (x, y) pt (x, y) Φ c , tα/β t1/β then either the Dirichlet form is local or E (α+β) Φ(s) (1 + s)− . ' (The symbol means that the ratio of the left hand side and right hand side is bounded between' two positive constants). Proof of (b). Let u be a non-constant function from L2 (M, µ). Choose a ball Q M where u is non-constant and let a > b be two real values such that the sets ⊂ A = x Q : u (x) > a and B = x Q : u (x) < b { ∈ } { ∈ } have positive measures.

B={u

A={u>a}

Figure 2.1: Sets A and B

D = diam Q then we have, for almost all x, y Q, ∈ 1 D pt (x, y) Φ , ≥ tα/β t1/β whence for any t > 0

1 2 (u) t (u) (u(x) u (y)) pt(x, y)dµ(y)dµ(x) E ≥ E ≥ 2t − ZA BZ 1 D (a b)2 µ (A) µ (B) Φ ≥ − 2t1+α/β t1/β c D = 0 Φ , t1+α/β t1/β 2.2. VOLUME OF BALLS 21

(α+β) where c0 > 0. If the inequality Φ (s) C (1 + s)− fails then there exists a ≤ sequence sk such that { } → ∞ α+β s Φ(sk) as k . k → ∞ → ∞

Deﬁne a sequence tk from the condition { } D s k = 1/β tk so that tk 0 as k . We have → → ∞ 1 D D (α+β)sα+β s k , 1+α/β Φ 1/β = − k Φ( k) as tk tk ! → ∞ → ∞ whence (u) = . Hence,E we have∞ arrived at the conclusion that the domain of the form may contain only constants. Since is dense in L2 (M, µ), this isF not possible,E which ﬁnishes the proof. F

2.2 Volume of balls

Denote by B (x, r) a metric ball in (M, d), that is

B(x, r) := y M : d(x, y) < r . { ∈ } Theorem 2.2. (AG, J.Hu, K.-S. Lau ’03) Let pt (x, y) be a heat kernel on (M, d, µ). Assume that it is stochastically complete and that it satisﬁes the two- sided estimate C d (x, y) pt (x, y) Φ c . ((1)) tα/β t1/β Then, for all x M and r > 0, ∈ µ(B(x, r)) rα, ' that is, µ is α-regular. α Consequently, dimH (M, d) = α and µ H on all Borel subsets of M, where Hα is the Hausdorﬀ measure of the dimension' α in M. In particular, the parameter α is the invariant of the metric space (M, d), and measure µ is determined (up to a factor 1) by the metric space (M, d). Proof. For all r, t > 0 and for almost' all x M we have ∈

pt(x, y)dµ(y) 1. ((2)) ≤ ZB(x,r) 22 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS

It follows from (2) that

1 − µ(B(x, r)) einf pt(x, y) . ≤ y B(x,r) ∈ 1 1/β Choose ε > 0 so that Φ (ε) > 0. Choosing t from the identity r = c− εt we obtain

C r α α einf pt(x, y) Φ c = c0r− ε Φ(ε) , y B(x,r) ≥ tα/β t1/β ∈ whence µ (B (x, r)) Crα. ((3)) ≤ To prove the lower bound for µ (B (x, r)), we ﬁrst show that for all 0 < t εrβ and almost all x M, ≤ ∈ 1 pt(x, y)dµ(y) , ((4)) M B(x,r) ≤ 2 Z \ k provided ε > 0 is suﬃciently small. Setting rk = 2 r and using the monotonicity of Φ and (3) we obtain

∞ pt(x, y)dµ(y) = pt (x, y) dµ(y) M B(x,r) k B(x,rk+1) B(x,rk) Z \ X=0 Z \ ∞ r Ct α/βΦ k dµ(y) − t1/β ≤ k B(x,rk+1) B(x,rk) X=0 Z \ ∞ α α/β rk Cr t− Φ ≤ k+1 t1/β k=0 X α ∞ 2kr 2kr = C Φ 0 t1/β t1/β k X=0 ∞ α ds C0 s Φ(s) . ((5)) ≤ 1 r/t1/β s Z 2 By Theorem 2.1(b), the integral (5) converges. Hence, its value can be made arbitrarily small provided rβ/t is large enough, whence (4) follows. By the stochastic completeness of the heat kernel and (4) we conclude that, under the condition 0 < t εrβ, ≤ 1 pt(x, y)dµ(y) , ≥ 2 ZB(x,r) whence 1 1 − µ(B(x, r)) esup pt(x, y) . ≥ 2 y B(x,r) ∈ ! 2.3. BESOV SPACES 23

Finally, choosing t = εrβ and using the upper bound

α/β α α/β pt(x, y) Ct− Φ(0) = Cr− ε− Φ (0) , ≤ we obtain µ (B (x, r)) crα, ≥ which ﬁnishes the proof.

2.3 Besov spaces

Fix α > 0, σ > 0 and introduce the following seminorms on L2 = L2 (M, µ):

1 N α,σ u u x u y 2 dµ x dµ y 2, ( ) = sup α+2σ ( ) ( ) ( ) ( ) ∞ 0

2 2 α,σ u α,σ = u + N (u). Λ2, 2 2, k k ∞ k k ∞ α,σ α,σ Similarly, one deﬁnes the space Λ2,2 . More generally one can deﬁne Λp,q for p [1, + ) and q [1, + ]. ∈ ∞ ∈ n ∞ In the case of R , we have the following relations

n,σ n σ n Λp,q (R ) = Bp,q (R ) , 0 < σ < 1, n,1 n 1 n Λ2, (R ) = Wp (R ) , ∞ n,1 n Λ2,2 (R ) = 0 , n,σ n { } Λp,q (R ) = 0 , σ > 1. { } σ 1 α,σ where Bp,q is the Besov space and Wp is the Sobolev space. The spaces Λp,q will also be called Besov spaces. Theorem 2.3. (Jonsson ’96, Pietruska-Paluba ’00, AG, J.Hu, K.-S. Lau ’03) Let pt be a heat kernel on (M, d, µ). Assume that it is stochastically complete and that it satisﬁes the following estimate for all t > 0 and almost all x, y M: ∈ 1 d(x, y) 1 d(x, y) Φ1 pt (x, y) Φ2 , ((6)) tα/β t1/β ≤ ≤ tα/β t1/β 24 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS

where α, β be positive constants, Φ1 and Φ2 monotone decreasing functions from [0, + ) to [0, + ) such that Φ1 (s) > 0 for some s > 0 and ∞ ∞

∞ α+β ds s Φ2(s) < . ((7)) s ∞ Z Then, for any u L2, ∈ α,β/2 (u) N2, (u), E ' ∞ α,β/2 and, consequently, = Λ2, . F ∞ By Theorem 2.1, the upper bound in (6) implies that either ( , ) is local or E F (α+β) Φ2 (s) c (1 + s)− . ≥ Since the latter contradicts (7), the form ( , ) must be local. For non-local forms E F β/2 the statement is not true. For example, for the operator ( ∆) in Rn we have β/2 n,β/2 β/2 n,β/− 2 = B2,2 = Λ2,2 that is strictly smaller than B2, = Λ2, . This case will be Fcovered by another theorem. ∞ ∞

Theorem 2.4. (St´os’00) Let pt be a stochastically complete heat kernel on (M, d, µ) satisfying estimate (6) with functions

(α+β) Φ1 (s) Φ2 (s) (1 + s)− . ' ' Then, for any u L2, ∈ α,β/2 (u) N (u) E ' 2,2 and, consequently, = Λα,β/2. F 2,2

2.4 Subordinated semigroup

Let be the generator of a heat semigroup Pt . Then for any δ (0, 1) the L δ { } ∈ t σ operator is also a generator of a heat semigroup, that is, the semigroup e− L is a heatL semigroup. Furthermore, there is the following relation between the two semigroups t δ ∞ s e− L = e− Lηt (s) ds Z0 where ηt (s) is a subordinator. Using the known estimates for ηt (s), one obtain the following result.

Theorem 2.5. Let a heat kernel pt satisfy the estimate (6) where Φ1 (s) > 0 for some s > 0 and ∞ ds α+β0 s Φ2 (s) < , s ∞ Z0 2.5. THE WALK DIMENSION 25

δ where β0 = δβ, 0 < δ < 1. Then the heat kernel qt (x, y) of operator satisﬁes the estimate L

(α+β0) 1 d (x, y) − t α/β0 qt (x, y) 1 + min t− , , ' tα/β0 t1/β0 ' α+β0 d (x, y) ! for all t > 0 and almost all x, y M. ∈

2.5 The walk dimension

It follows from deﬁnition that the Besov seminorm 1 N α,σ u u x u y 2 dµ x dµ y 2, ( ) := sup α+2σ ( ) ( ) ( ) ( ) ∞ 0

α,σ 2 α,σ Λ2, := u L : N2, (u) < ∞ ∈ ∞ ∞ shrinks. For a certain value of σ this space can become trivial. For example, as was n,σ n n,σ n already mentioned, Λ2, (R ) = 0 for σ > 1, while Λ2, (R ) is non-trivial for σ 1. ∞ { } ∞ ≤ Deﬁnition. Fix α > 0 and set

α,β/2 2 β∗ := sup β > 0 : Λ2, is dense in L (M, µ) . ((8)) ∞ n o α,β/2 The number β∗ [0, + ] is called the critical exponent of the family Λ2, ∈ ∞ ∞ β>0 of Besov spaces. n o

Note that the value of β∗ is an intrinsic property of the space (M, d, µ), which is deﬁned independently of any heat kernel. For example, for Rn with α = n we have β∗ = 2. Theorem 2.6. (A.Jonsson ’96, K.Pietruska-Paluba ’00, AG, J.Hu, K.-S. Lau ’03) Let pt be a heat kernel on a metric measure space (M, d, µ). If the heat kernel is stochastically complete and satisﬁes (6), that is, 1 d(x, y) 1 d(x, y) Φ1 pt (x, y) Φ2 , (copy of (6)) tα/β t1/β ≤ ≤ tα/β t1/β where Φ1 (s) > 0 for some s > 0 and

∞ α+β+ε ds s Φ2(s) < ((9)) s ∞ Z for some ε > 0, then β = β∗. 26 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS

By Theorem 2.1, (9) implies that the Dirichlet form ( , ) is local. For non-local n E F forms the statement is not true: for example, in R for symmetric stable processes we have β < 2 = β∗. Corollary 2.7. Under the hypotheses of Theorem 2.6, the values of the parameters α and β are the invariants of the metric space (M, d) alone. Moreover, we have

α α,β/2 µ H and N2, . ' E' ∞ Consequently, both measure µ and the energy form are determined (up to a factor 1) by the metric space (M, d) alone. E ' Example. Consider in Rn the Gauss-Weierstrass heat kernel

1 x y 2 p x, y t ( ) = n/2 exp | − | (4πt) − 4t ! and its generator = ∆ in L2 (Rn) with the Lebesgue measure. Then α = n, β = 2, and L − (u) = u 2 dx. E n |∇ | ZR Consider now another elliptic operator in Rn: n 1 ∂ ∂ = aij (x) , m (x) ∂xi ∂xj L − i,j=1 X where m (x) and aij (x) are continuous functions, m (x) > 0 and the matrix (aij (x)) is positive deﬁnite. The operator is symmetric with respect to measure L dµ = m (x) dx, and its Dirichlet form is ∂u ∂u (u) = aij (x) dx. E n ∂xi ∂xj ZR

Let d (x, y) = x y and assume that the heat kernel pt (x, y) of satisﬁes the conditions of Theorem| − | 2.6. Then we conclude by Corollary 2.7 that Lα and β must be the same as for the Gauss-Weierstrass heat kernel, that is, α = n and β = 2; moreover, measure µ must be comparable to the Lebesgue measure, which implies that m 1, and the energy form must admit the estimate ' (u) u 2 dx, E ' n |∇ | ZR which implies that the matrix (aij (x)) is uniformly elliptic. Hence, the operator is uniformly elliptic. L 2.6. INEQUALITIES FOR THE WALK DIMENSION 27

By Aronson’s theorem the heat kernel for uniformly elliptic operators satisﬁes the estimate C x y 2 p x, y c . t ( ) n/2 exp | − | ' t − t ! What we have proved implies the converse to Aronson’s theorem: if the Aronson estimate holds for the operator then is uniformly elliptic. L L The next theorem handles the non-local case.

Theorem 2.8. Let pt be a heat kernel on a metric measure space (M, d, µ). If the heat kernel satisﬁes the lower bound 1 d (x, y) pt (x, y) Φ1 , ≥ tα/β t1/β where Φ1 (s) > 0 for some s > 0, then β β∗. ≤ Proof. In the proof of Theorem 2.3 one shows that the lower bound of the heat α,β/2 kernel implies Λ2, (and the opposite inclusion follows from the upper bound F ⊂ ∞ 2 and stochastic completeness). Since is dense in L , it follows that β β∗. F ≤ 2.6 Inequalities for the walk dimension

Deﬁnition. We say that a metric space (M, d) satisﬁes the chain condition if there exists a (large) constant C such that for any two points x, y M and for any positive n ∈ integer n there exists a sequence xi i=0 of points in M such that x0 = x, xn = y, and { } d(x, y) d(xi, xi+1) C , for all i = 0, 1, ..., n 1. ((10)) ≤ n − n The sequence xi is referred to as a chain connecting x and y. { }i=0 Theorem 2.9. (AG, J.Hu, K.-S. Lau ’03) Let (M, d, µ) be a metric measure space and assume that µ(B(x, r)) rα ((11)) ' for all x M and 0 < r 1. Then ∈ ≤

β∗ 2. ≥ If in addition (M, d) satisﬁes the chain condition then

β∗ α + 1. ≤

Observe that the chain condition is essential for the inequality β∗ α + 1 to be true. Indeed, assume for a moment that the claim of Theorem 2.9 holds≤ without the 1/γ chain condition, and consider a new metric d0 on M given by d0 = d where γ > 1. 28 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS

Let us mark by a dash all notions related to the space (M, d0, µ) as opposed to those of (M, d, µ). It is easy to see that α0 = αγ and β∗0 = β∗γ. Hence, if Theorem 2.9 could apply to the space (M, d0, µ) it would yield β∗γ αγ +1 which implies β∗ α ≤ ≤ because γ may be taken arbitrarily large. However, there are spaces with β∗ > α, for example SG. Clearly, the metric d0 does not satisfy the chain condition; indeed the inequality (10) implies d0(x, y) d0(xi, xi+1) C , ≤ n1/γ which is not good enough. Note that if in the inequality (10) we replace n by n1/γ then the proof of Theorem 2.9 will give β∗ α + γ instead of β∗ α + 1. ≤ α,1 ≤ 2 Proof. To prove that β∗ 2, it suﬃces to show that Λ2, is dense in L . Let u be a Lipschitz function with≥ a bounded support A and let∞ L be the Lipschitz constant of u. Then, for any r (0, 1], ∈ 1 (u(x) u(y))2 dµ(y)dµ(x) rα+2 − ZM ZB(x,r) 1 Lr2dµ(y)dµ(x) ≤ rα+2 ZA1 ZB(x,r) Cµ (A1) , ≤ where Ar denotes the closed r-neighborhood of A. It follows that

α,1 N2, (u) Cµ (A1) < , ∞ ≤ ∞ α,1 whence we conclude that u Λ2, . Let now A be any bounded∈ ∞ closed subset of M. For any positive integer n, consider the function on M

fn (x) = (1 nd (x, A)) , − + α,1 which is Lipschitz and is supported in A1/n. Hence, fn Λ2, . Clearly, fn 1A ∈ ∞ → 2 α,1 in L as n , whence it follows that 1A Λ2, , where the bar means closure in L2. Since the→ ∞ linear combinations of the indicator∈ ∞ functions of bounded closed sets 2 α,1 2 form a dense subset in L , it follows that Λ2, = L , which was to be proved. ∞ Now let us prove that β∗ α + 1 assuming the chain condition. The hypothesis ≤2 (11) implies that the space L (M, µ) is -dimensional. The inequality β∗ α + 1 ∞ α,β/2 ≤ will be proved if we show that, for any β > α + 1, the space Λ2, contains only α,β/2 ∞ constants, that is, N2, (u) < implies u const. ∞α,β/2 ∞ ≡ By deﬁnition of N2, we have, for any 0 < r 1, ∞ ≤

α,β/2 α β 2 N2, (u) r− − u(x) u(y) dµ(y)dµ(x). ∞ ≥ | − | d(ZZx,y)

l Fix some 0 < r 1 and assume that we have a sequence of disjoint balls Bk k=0 of the same radius≤ 0 < ρ < 1, such that for all k = 0, 1, ..., l 1 { } −

dmax (Bk,Bk+1) := sup d(x, y): x Bk, y Bk+1 < r. ((12)) { ∈ ∈ } Then we have

l 1 α,β/2 α β − 2 N2, (u) r− − u(x) u(y) dµ(y)dµ(x). ∞ ≥ k Bk Bk+1 | − | X=0 Z Z Now use the following notation 1 u := u dµ A µ(A) ZA and observe that, for any two measurable sets A, B M of ﬁnite measure, the following inequality takes place ⊂

2 2 u(x) u(y) dµ(x)dµ(y) µ(A)µ(B) uA uB , | − | ≥ | − | ZA ZB which is proved by a straightforward computation. It follows that

l 1 α,β/2 α β − 2 N2, (u) r− − µ (Bk) µ (Bk+1) uBk uBk+1 ∞ ≥ k − X=0 l 1 − α β 2α 2 cr− − ρ uB uB ≥ k − k+1 k=0 X l 1 2 − α β 2α 1 cr− − ρ uB uB ≥ l k − k+1 k=0 ! X α β 2α 1 2 = cr− − ρ (uB uB ) . l 0 − l

Now we construct such a sequence Bk . Fix two distinct points x, y M and recall that, by the chain condition, for any{ positive} integer n there exists a∈ sequence n of points xi such that x0 = x, xn = y, and { }i=0 d(x, y) d(xi, xi+1) < C := ρ, for all 0 i < n. n ≤

l It is possible to show that, for large enough n, there exists a subsequence xi { k }k=0 such that xi = x, xi = y, the balls B (xi , ρ) are disjoint, and 0 l { k }

d(xik , xik+1 ) < 5ρ, 30 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS

for all k = 0, 1, ..., l 1. Setting Bk := B (xi , ρ) and noticing that − k

dmax (Bk,Bk+1) < 5ρ + 2ρ = 7ρ =: r and that r < 1 provided n is large enough, we obtain

2 2 uB(x,ρ) uB(y,ρ) = (uB uB ) − 0 − l α,β/2 β+α 2α CN2, (u) r ρ− l ≤ ∞ α,β/2 β α CN2, (u) ρ − n ≤ ∞ α,β/2 β α 1 CN2, (u) ρ − − d (x, y) . ≤ ∞ By the Lebesgue theorem, we have, for almost all x M, ∈

lim uB(x,ρ) = u(x), ρ 0 → Letting n (that is ρ 0) we obtain, for almost all x, y M, → ∞ → ∈

2 α,β/2 β α 1 (u(x) u(y)) CN2, (u) lim ρ − − d (x, y) = 0. − ≤ ∞ ρ 0 → α,β/2 Since N2, (u) < and β > α + 1, the above limit is equal to 0 whence u const. ∞ ∞ ≡

Corollary 2.10. AG, J.Hu, K.-S. Lau ’03) Let pt be a stochastically complete heat kernel on (M, d, µ) such that

C d (x, y) pt (x, y) Φ c . tα/β t1/β (a) If for some ε > 0 ∞ ds sα+β+εΦ(s) < ((11)) s ∞ Z then β 2. ≥ (b) If (M, d) satisﬁes the chain condition then β α + 1. ≤ Proof. By Theorem 2.2 µ is α-regular so that Theorem 2.9 applies. (a) By Theorem 2.9 β∗ 2 and by Theorem 2.6, β = β∗, whence β 2. ≥ ≥ (b) By Theorem 2.9 β∗ α + 1 and by Theorem 2.8 β β∗, whence β α + 1. ≤ ≤ ≤ Note that the condition (11) can occur only for a local Dirichlet form . If both (11) and the chain condition are satisﬁed then we obtain E

2 β α + 1. ((12)) ≤ ≤ 2.6. INEQUALITIES FOR THE WALK DIMENSION 31

1 2 3 4 α

Figure 2.2: The set 2 β α + 1 ≤ ≤

This inequality was stated by M.Barlow ’98 without proof. The set of couples (α, β) satisfying (12) is shown on the diagram: Barlow ’04 proved that any couple of α, β satisfying (12) can be realized for the heat kernel estimate

1 β β 1 C d (x, y) − pt (x, y) exp c ((13)) tα/β − t ! For a non-local form, we can only claim that

0 < β α + 1 ≤ (under the chain condition). In fact, any couple α, β in the range

0 < β < α + 1 can be realized for the estimate

(α+β0) 1 d (x, y) − pt (x, y) 1 + . ' tα/β0 t1/β0 Indeed, if is the generator of a diﬀusion with parameters α and β satisfying (13) then the operatorL δ, δ (0, 1), generates a jump process with the walk dimension L ∈ β0 = δβ and the same α (cf. Theorem 2.5). Clearly, β0 can take any value from (0, α + 1). It is not known whether the walk dimension for a non-local form can be equal to α + 1. 32 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS 2.7 Identifying Φ in the local case

Theorem 2.11. (AG, T.Kumagai ’09) Assume that the metric space (M, d) satis- ﬁes the chain condition and all metric balls are precompact. Let pt be a stochastically complete heat kernel in (M, d, µ). Assume that the associated Dirichlet form ( , ) is regular, and the following estimate holds with some α, β > 0 and Φ : [0, + E)F [0, + ): ∞ → ∞ C d (x, y) pt (x, y) Φ c . tα/β t1/β Then the following dichotomy holds:

β either the Dirichlet form is local, 2 β α+1, and Φ(s) C exp( cs β 1 ). • E ≤ ≤ − − (α+β) or the Dirichlet form is non-local, β α + 1, and Φ(s) (1 + s)− . • E ≤ ' Proof. The non-local case as well as the inequality β α + 1 follow from Corollary 2.10 and Theorem 2.1. We only need to treat the local≤ case: to show that β β 1 β 2 and Φ (s) C exp( cs − ). ≥Integrating the heat kernel− over B (x, r)c we obtain as in the proof of Theorem 2.2 ∞ α ds pt (x, y) dµ (y) C s Φ(s) , B(x,r)c ≤ 1 r/t1/β s Z Z 2 α β where the integral converges by Theorem 2.1(b): Φ (s) (1 + s)− − . Therefore, for any ε > 0 there is K > 0 such that the following estimate≤ holds

pt (x, y) dµ (y) ε ((14)) c ≤ ZB(x,r) for almost all x M, whenever r Kt1/β. The estimate∈ (14) allows self-improvement≥ similarly to bootstrapping argument of Barlow for Px τ B(x,r) t ε. ≤ ≤ Technically we use a modiﬁcation of the method of Hebisch and Saloﬀ-Coste ’01 (here the regularity of the Dirichlet form is used) and obtain the following: for all t, r, λ > 0 1/β pt (x, y) dµ (y) C exp λt crλ ((15)) B(x,r)c ≤ − Z (AG, J.Hu ’08 and ’10). If β < 1 then letting in (15) λ , we obtain that the right hand side in (15) goes to 0. Letting then r 0, we→ obtain ∞ that, for almost all x M, → ∈

pt (x, y) dµ (y) = 0, M x Z \{ } 2.7. IDENTIFYING Φ IN THE LOCAL CASE 33 which is not possible under the present hypotheses. This contradiction proves that β 1. ≥Setting in (15) β cr β 1 − , if β > 1, λ =  2t 1  t− , if β = 1 we obtain that, for all positive r, t and almost all x M,  ∈ 1 β β 1 r − C exp c t , if β > 1 pt (x, y) dµ (y) − c ≤  r ZB(x,r) C exp c , if β = 1  − t Using the semigroup identity, we have, for all t > 0, almost all x, y M, and 1 ∈ r := 2 d (x, y),

pt (x, y) = p t (x, z) p t (z, y) dµ(z) 2 2 ZM + p t (x, z) p t (z, y) dµ(z) ≤ c c 2 2 ZB(x,r) ZB(y,r)

esup p t (z, y) p t (x, z) dµ(z) 2 2 ≤ z M B(x,r)c ∈ Z

+ esup p t (x, z) p t (y, z) dµ(z). 2 2 z M B(y,r)c ∈ Z α/β Using esup pt Ct− , we obtain, for almost all x, y M, ≤ ∈ 1 β β 1 C d (x, y) − exp c , if β > 1  tα/β − t pt (x, y) ! ≤   C r  exp c , if β = 1 tα − t  It follows that pt satisﬁes the two-sided estimate (6) with functions

Φ1 (s) := CΦ(cs) and β β 1 C exp cs − , if β > 1, Φ2 (s) := − ( C exp ( cs) , if β = 1. − Then Theorem 2.6 applies and yields β = β∗. By Theorems 2.2 and 2.9 we have β∗ 2 whence β 2. ≥Consequently,≥ we obtain the upper bound

1 β β 1 C d (x, y) − pt (x, y) exp c . ≤ tα/β − t ! 34 CHAPTER 2. CONSEQUENCES OF HEAT KERNEL BOUNDS

By hypothesis we have the lower bound

α/β 1/β pt (x, y) ct− provided d (x, y) st ≥ ≤ where s > 0 is such that Φ (s) > 0. The standard chaining argument using the chain condition yields then

1 β β 1 C d (x, y) − pt (x, y) exp c ≥ tα/β − t ! for almost all x, y. Combining with the upper bound, we obtain

β Φ(s) C exp( cs β 1 ), − − which ﬁnishes the proof. Chapter 3

Upper bounds of the heat kernel

3.1 Ultracontractive semigroups

Let (M, d, µ) be a metric measure space and ( , ) be a Dirichlet form in L2 (M, µ) E F t and Pt be the associated heat semigroup, Pt = e− L where is the generator of { } L ( , ). The question to be discussed here is whether Pt possesses the heat kernel, E F that is, a function pt (x, y) that is non-negative, jointly measurable in (x, y), and satisﬁes the identity

Ptf (x) = pt (x, y) f (y) dµ (y) ZM for all f L2, t > 0, and almost all x M. Usually the conditions that ensure the existence∈ of the heat kernel give at the∈ same token some upper bounds. p q Given two parameters p, q [0, + ], deﬁne the L L norm of Pt by ∈ ∞ →

Ptf q Pt Lp Lq = sup k k . k k → f Lp L2 0 f p ∈ ∩ \{ } k k p In fact, the Markovian property allows to extend Pt to an operator in L so that the range Lp L2 of f can be replaced by Lp. Also, it follows from the Markovian ∩ property that Pt Lp Lp 1 for any p. k k → ≤ p q Deﬁnition. The semigroup Pt is said to be L L ultracontractive if there exists a positive decreasing function{ } γ on (0, + ),→ called the rate function, such that, for each t > 0 ∞ Pt Lp Lq γ (t) . k k → ≤

p q q By the symmetry of Pt, if Pt is L L ultracontractive, then Pt is also L ∗ p → → L ∗ ultracontractive with the same rate function, where p∗ and q∗ are the H¨older 1 2 conjugates to p and q, respectively. In particular, Pt is L L ultracontractive if 2 → and only if it is L L∞ ultracontractive. → Theorem 3.1.

35 36 CHAPTER 3. UPPER BOUNDS OF THE HEAT KERNEL

1 2 (a) The heat semigroup Pt is L L ultracontractive with a rate function γ, { } → if and only if Pt has the heat kernel pt satisfying the estimate { } 2 esup pt (x, y) γ (t/2) for all t > 0 x,y M ≤ ∈

1 (b) The heat semigroup Pt is L L∞ ultracontractive with a rate function γ, { } → if and only if Pt has the heat kernel pt satisfying the estimate { }

esup pt (x, y) γ (t) for all t > 0. x,y M ≤ ∈

This result is “well-known” and can be found in many sources. However, there are hardly complete proofs of the measurability of the function pt (x, y) in (x, y), which is necessary for many applications, for example, to use Fubini. Normally the existence of the heat kernel is proved in some speciﬁc setting where pt (x, y) is 2 continuous in (x, y), or one proves just the existence of a family of functions pt,x L so that ∈

Ptf (x) = (pt,x, f) = pt,x (y) f (y) dµ (y) ZM for all t > 0 and almost all x. However, if one deﬁnes pt (x, y) = pt,x (y), then this function does not have to be jointly measurable. The proof of the existence of a jointly measurable version can be found in a preprint AG, J.Hu “Upper bounds of heat kernels on doubling spaces”, 2010. Most of the material of this chapter can also be found there.

3.2 Restriction of the Dirichlet form

Let Ω be an open subset of M. Deﬁne the function space Ω by F

F Ω = f : supp f Ω . F { ∈ F ⊂ } 2 Clearly, Ω is a closed subspace of and a subspace of L (Ω). F F 2 Lemma 3.2. If ( , ) is a regular Dirichlet form in L (M) then ( , Ω) is a E F 2 E F regular Dirichlet form in L (Ω). If ( , ) is (strongly) local then so is ( , Ω). E F 2E F The regularity is used, in particular, to ensure that Ω is dense in L (Ω). From now on let us assume that ( , ) is a regular Dirichlet form.F Other consequences of this assumptions are as follows:E F

1. The existence of cutoﬀ functions: for any compact set K and any open set U K, there is a function ϕ C0 (U) such that 0 ϕ 1 and ϕ 1 in an⊃ open neighborhood of K. ∈ F ∩ ≤ ≤ ≡

2. The existence of a Hunt process Xt t 0 , Px x M associated with ( , ). { } ≥ { } ∈ E F 3.3. FABER-KRAHN AND NASH INEQUALITIES 37

(see Fukushima, Oshima, Takeda “Dirichlet forms and symmetric Markov processes”). Hence, for any open subset Ω M, we have the Dirichlet form ( , Ω) that is called a restriction of ( , ) to Ω.⊂ E F E F Example. Consider in Rn the canonical Dirichlet form

(u) = u 2 dx E n |∇ | ZR W 1 1 n 1 2 1 in = W2 (R ). Then Ω = C0 (Ω) =: H0 (Ω) . F F Using the restricted form ( , Ω) corresponds to imposing the Dirichlet boundary c E F conditions on ∂Ω (or on Ω ), so that the form ( , Ω) could be called the Dirichlet form with the Dirichlet boundary condition. E F Denote by Ω the generator of ( , Ω) and set L E F (u) λmin (Ω) := inf spec Ω = inf E 2 . ((1)) L u Ω 0 u ∈F \{ } k k2 Clearly, λmin (Ω) 0 and λmin (Ω) is decreasing when Ω expands. ≥ Example. If ( , ) is the canonical Dirichlet form in Rn and Ω is the bounded n E F domain in R then the operator Ω has the discrete spectrum λ1 (Ω) λ2 (Ω) L ≤ ≤ λ3 (Ω) ... that coincides with the eigenvalues of the Dirichlet problem ≤ ∆u + λu = 0 u ∂Ω = 0, | so that λ1 (Ω) = λmin (Ω).

3.3 Faber-Krahn and Nash inequalities

Continuing the above example, we have by a theorem of Faber-Krahn

λ1 (Ω) λ1 (Ω∗) ≥ where Ω∗ is the ball of the same volume as Ω. If r is the radius of Ω∗ then we have

c0 c c λ1 (Ω∗) = = = r2 Ω 2/n Ω 2/n | ∗| | | whence 2/n λ1 (Ω) cn Ω − . ≥ | | It turns out that this inequality, that we call the Faber-Krahn inequality, is intimately related to the existence of the heat kernel and its upper bound. Theorem 3.3. Let ( , ) be a regular Dirichlet form in L2 (M, µ). Fix some constant ν > 0. Then theE followingF conditions are equivalent: 38 CHAPTER 3. UPPER BOUNDS OF THE HEAT KERNEL

(i) ( The Faber-Krahn inequality) There is a constant a > 0 such that, for all non-empty open sets Ω M, ⊂ ν λmin (Ω) aµ (Ω)− . ((2)) ≥ (ii) ( The Nash inequality) There exists a constant b > 0 such that

2+2ν 2ν (u) b u u − , ((3)) E ≥ k k2 k k1 for any function u 0 . ∈ F \ { } (iii) ( On-diagonal estimate of the heat kernel) The heat kernel exists and satisﬁes the upper bound 1/ν esup pt (x, y) ct− ((4)) x,y M ≤ ∈ for some constant c and for all t > 0.

The relation between the parameters a, b, c is as follows:

ν a b c− ' ' where the ratio of any two of these parameters is bounded by constants depending only on ν. In Rn ν = 2/n. (ii) (iii) Nash ’58 (iii)⇒ (ii) Carlen-Kusuoka-Stroock ’87 (i) ⇒(iii) AG ’94, Carron ’94. Proof⇔ of (i) (ii) (iii). Observe ﬁrst that (ii) (i) is trivial: choosing in ⇒ ⇒ ⇒ (3) a function u Ω 0 and applying the Cauchy-Schwarz inequality ∈ F \{ } u µ (Ω)1/2 u , k k1 ≤ k k2 we obtain ν 2 (u) bµ (Ω)− u , E ≥ k k2 whence (2) follow by the variational principle (1). The opposite inequality (i) (ii) is a bit more involved, and we prove it for ⇒ functions 0 u C0 (M) (a general u requires some approximation ≤ ∈ F ∩ ∈ F argument). By the Markovian property, we have (u t)+ C0 (M) for any t > 0 and − ∈ F ∩ (u) (u t) . E ≥ E − + Consider for any s > 0 the set

Us := x M : u (x) > s , { ∈ } 3.3. FABER-KRAHN AND NASH INEQUALITIES 39

which is clearly open and precompact. If t > s then (u t) is supported in Us, − + whence (u t) U . It follows from (1) − + ∈ F s

2 (u t) λmin (Us) (u t) dµ. E − + ≥ − + ZUs 2 Set for simplicity A = u 1 and B = u . Since u 0, we have k k k k2 ≥ (u t)2 u2 2tu, − + ≥ − which implies that

(u t)2 dµ = (u t)2 dµ B 2tA. − + − + ≥ − ZUs ZM On the other hand, we have 1 A µ(Us) u dµ , ≤ s ≤ s ZUs which together with the Faber-Krahn inequality implies

ν ν s λmin (Us) aµ (Us)− a . ≥ ≥ A Combining the above lines, we obtain

ν 2 s (u) λmin (Us) (u t)+ dµ a (B 2tA) . E ≥ Us − ≥ A − Z Letting t s+ and then choosing s = B , we obtain → 4A ν ν s B B a ν+1 2ν (u) a (B 2sA) = a = B A− , E ≥ A − 4A2 2 4ν2 which is exactly (3) . 2 1 t To prove (ii) (iii), choose f L L , and consider u = Ptf. Since u = e− Lf and d u = u,⇒ we have ∈ ∩ dt −L d 2 d 2+2ν 2ν 2+2ν 2ν u = (u, u) = 2 ( u, u) = 2 (u, u) 2b u u − 2b u f − , dt k k2 dt − L − E ≤ − k k2 k k1 ≤ − k k2 k k1 since u f . Solving this diﬀerential inequality, we obtain k k1 ≤ k k1 2 1/v 2 Ptf ct− f , k k2 ≤ k k1 1 2 that is, the semigroup Pt is L L ultracontractive with the rate function γ (t) = 1/v → √ct− . By Theorem 3.1 we conclude that the heat kernel exists and satisﬁes (4). 40 CHAPTER 3. UPPER BOUNDS OF THE HEAT KERNEL

Let M be a Riemannian manifold with the geodesic distance d and the Rieman- nian volume µ. Let ( , ) be the canonical Dirichlet form on M. The heat kernel on manifolds always existsE F and is a smooth function. In this case the estimate (4) is equivalent to the on-diagonal upper bound 1/ν sup pt (x, x) ct− . x M ≤ ∈ It is known (but non-trivial) that the on-diagonal estimate implies the Gaussian upper bound 2 1/ν d (x, y) pt (x, y) Ct− exp , ≤ −(4 + ε) t for all t > 0 and x, y M, which is due to the speciﬁc property of the geodesic distance function that ∈ d 1. More about heat kernels|∇ | ≤ on manifolds can be found in

Figure 3.1:

In the context of abstract metric measure space the distance function does not have to satisfy this property and typically it does not (say, on fractals). Conse- quently, one needs some additional conditions that would relate the distance function to the Dirichlet form and imply the oﬀ-diagonal bounds.

3.4 Oﬀ-diagonal upper bounds

From now on let ( , ) be a regular local Dirichlet form, so that the associated E F Hunt process Xt t 0 , Px x M is a diﬀusion. Recall that it is related to the heat { } ≥ { } ∈ 3.4. OFF-DIAGONAL UPPER BOUNDS 41

semigroup Pt of ( , ) by means of the identity { } E F

Ex (f (Xt)) = Ptf (x)

for all f b (M), t > 0 and almost all x M. Fix two∈ B parameters α > 0 and β > 1 and∈ introduce some conditions.

(Vα) (Volume regularity) For all x M and r > 0, ∈ µ (B (x, r)) rα. '

(FK) (The Faber-Krahn inequality) For any open set Ω M, ⊂ β/α λmin (Ω) cµ (Ω)− . ≥

For any open set Ω M deﬁne the ﬁrst exist time from Ω by ⊂

τ Ω = inf t > 0 : Xt / Ω . { ∈ }

Xτ Xt

Figure 3.2: First exit time τ

A set N M is called properly exceptional, if it is a Borel set of measure 0 that ⊂ is almost never hit by the process Xt starting outside N. In the next conditions N denotes some properly exceptional set.

(Eβ)(An estimate for the mean exit time from balls) For all x M N and r > 0 ∈ \ β Exτ B(x,r) r ' (the parameter β is called the walk dimension of the process). 42 CHAPTER 3. UPPER BOUNDS OF THE HEAT KERNEL

(EΩ) (An isoperimetric estimate for the mean exit time) For any open subset Ω M, ⊂ β/α sup Ex (τ Ω) Cµ (Ω) . x Ω N ≤ ∈ \

If both (Vα) and (Eβ) are satisﬁed then we obtain for any ball B M ⊂ β β/α sup Ex (τ B) r µ (B) . x B N ' ' ∈ \ It follows that the balls are in some sense optimal sets for the condition (EΩ). n Example. If Xt is Brownian motion in R then it is known that

2 Exτ B(x,r) = cnr

so that (Eβ) holds which satisﬁes with β = 2. This can also be rewritten in the form

2/n Exτ B = cn B | | where B = B (x, r). It is also known that for any open set Ω Rn with ﬁnite volume and for any x Ω ⊂ ∈ Ex (τ Ω) Ex τ B(x,r) , ≤ provided ball B (x, r) has the same volume as Ω; that is, for a ﬁxed value of Ω , the mean exist time is maximal when Ω is a ball and x is its center. It follows that| |

2/n Ex (τ Ω) cn Ω ≤ | | so that (EΩ) is satisﬁed with β = 2 and α = n. Finally, introduce notation for the following estimates of the heat kernel:

(UE)(Sub-Gaussian upper estimate) The heat kernel exists and satisﬁes the estimate

1 β β 1 C d (x, y) − pt (x, y) exp c ≤ tα/β − t ! for all t > 0 and almost all x, y M. ∈ (ΦUE) (Φ-upper estimate) The heat kernel exists and satisﬁes the estimate 1 d (x, y) pt (x, y) Φ ≤ tα/β t1/β for all t > 0 and almost all x, y M, where Φ is a decreasing positive function on [0, + ) such that ∈ ∞ ∞ ds sαΦ(s) < . s ∞ Z0 3.4. OFF-DIAGONAL UPPER BOUNDS 43

(DUE)(On-diagonal upper estimate) The heat kernel exists and satisﬁes the estimate C pt (x, y) ≤ tα/β for all t > 0 and almost all x, y M. ∈ Clearly, (UE) (ΦUE) (DUE) . ⇒ ⇒ Theorem 3.4. Let (M, d, µ) be a metric measure space and let (Vα) hold. Let ( , ) be a regular, local, conservative Dirichlet form in L2(M, µ). Then, the fol- lowingE F equivalences are true:

(UE) (ΦUE) ⇔ (FK) + (Eβ) ⇔ (EΩ) + (Eβ) ⇔

Let us emphasize the equivalence

(UE) (EΩ) + (Eβ) ⇔ where the right hand side means the following: the mean exit time from all sets Ω satisﬁes the isoperimetric inequality, and this inequality is optimal for balls (up to a constant multiple). Note that the latter condition relates the properties of the diﬀusion (and, hence, of the Dirichlet form) to the distance function. Conjecture. Under the hypotheses of Theorem 3.4,

β (UE) (FK) + λmin (Br) r− ⇔ ' Indeed, the Faber-Krahn inequality (FK) can be regarded as an isoperimetric inequality for λmin (Ω), and the condition

β λmin (Br) r− ' means that (FK) is optimal for balls (up to a constant multiple). Theorem 3.4 is an oversimpliﬁed version of a result of AG, J.Hu ’10 where instead of (Vα) one uses the volume doubling condition, and other hypotheses must be appropriately changed. The following lemma is used in the proof of Theorem 3.4. Lemma 3.5. For any open set Ω M ⊂ 1 λmin (Ω) . ≥ esupx Ω Ex (τ Ω) ∈ 44 CHAPTER 3. UPPER BOUNDS OF THE HEAT KERNEL

Proof. Let GΩ be the Green operator in Ω, that is,

∞ 1 t Ω GΩ = − = e− L dt. LΩ Z0 We claim that Ex (τ Ω) = GΩ1 (x) for almost all x Ω. We have ∈ ∞ t Ω GΩ1 (x) = e− L 1Ω (x) dt Z0 ∞ Ω = Ex 1Ω Xt Z0 ∞ = Ex 1 t<τ Ω dt { } Z0 ∞ = Ex 1 t<τ Ω dt { } Z0 = Ex (τ Ω).

Setting m = esup Ex (τ Ω) x Ω ∈ 1 we obtain that GΩ1 m, so that m− GΩ is a Markovian operator. Therefore, 1 ≤ 1 m− GΩ L2 L2 1 whence spec GΩ [0, m]. It follows that spec Ω [m− , ) k k → ≤ 1 ∈ L ⊂ ∞ and λmin (Ω) m− . Proof of≥ Theorem 3.4 ‘ ’. We have the implications ⇐ (EΩ) L.3.5 (FK) T.3.3 (DUE) . ⇒ ⇒ In particular, we see that the heat kernel exists under any of the hypotheses of Theorem 3.4. The next observation is that

(Eβ) Px τ B(x,r) t ε ((5)) ⇒ ≤ ≤ for some ε (0, 1) provided r Kt1/β (like in Barlow’s lectures), which in turn yields ∈ ≥

pt (x, y) dµ (y) ε. ((6)) c ≤ ZB(x,r) It is easy to see that also (ΦUE) (6) just by direct integration as in the proof of Theorem 2.2. ⇒ 3.4. OFF-DIAGONAL UPPER BOUNDS 45

The condition (6) implies by bootstrapping

1 β β 1 r − pt (x, y) dµ (y) C exp c ((7)) c ≤ − t ZB(x,r) ! for all t, r > 0 and almost all x M, as it was mentioned in the proof of Theorem 2.11. ∈ Hence, any set of the hypothesis of Theorem 3.4 imply both (DUE) and (7). By an argument in the proof of Theorem 2.11 we conclude

(DUE) + (7) (UE) . ⇒ 46 CHAPTER 3. UPPER BOUNDS OF THE HEAT KERNEL Chapter 4

Two-sided bounds of the heat kernel

4.1 Using elliptic Harnack inequality

Now we would like to extend the results of Ch.3 to obtain also the lower estimates of the heat kernel. As before, (M, d, µ) is a metric measure space, and assume in addition that all metric balls are precompact. Let ( , ) is a local regular conservative Dirichlet form in L2 (M, µ). E F Deﬁnition. We say that a function u is harmonic in an open set Ω M if ∈ F ⊂ (u, v) = 0 for all v (Ω) . E ∈ F

For example, if M = Rn and ( , ) is the canonical Dirichlet form in Rn then E F 1 n we obtain the following definition: a function u W2 (R ) is harmonic in an open n ∈ set Ω R if ⊂ u, v dx = 0 n h∇ ∇ i ZR 1 for all v H0 (Ω) v C0∞ (Ω). This of course implies that ∆u = 0 in a weak sense in Ω∈ and, hence,⇔ u ∈is harmonic in Ω in the classical sense. However, unlike the 1 n classical definition, we a priori require u W2 (R ) . ∈ Definition. We say that M satisfies the elliptic Harnack inequality (H) if there exist constants C > 1 and δ (0, 1) such that for any ball B (x, r) and for any function u that is non-negative∈ and harmonic in B (x, r), ∈ F esup u C einf u. B(x,δr) ≤ B(x,δr)

Theorem 3.6. (AG, A.Telcs ’10) If the hypotheses (Vα)+(Eβ)+(H) are satisfied, then the heat kernel pt (x, y) exists, is Höldercontinuous in x, y M, and satisfies for all t > 0 and all x, y M the following estimates: ∈ ∈ 47 48 CHAPTER 4. TWO-SIDED BOUNDS OF THE HEAT KERNEL

(UE) a sub-Gaussian upper estimate

1 β β 1 C d (x, y) − pt (x, y) exp c , ≤ tα/β − t !

(NLE) and the near-diagonal lower estimate

c 1/β pt (x, y) provided d (x, y) ηt , ≥ tα/β ≤ where η > 0 is a small enough constant.

Furthermore, we have the equivalence

(Vα) + (UE) + (NLE) (Vα) + (Eβ) + (H) . ⇔

This theorem is proved in AG, A.Telcs ’10 in a more general setting of volume doubling instead of (Vα). Approach to the proof. First one shows that (Vα) + (Eβ) + (H) (FK), ⇒ which is quite involved and uses, in particular, Lemma 3.5. Having (Vα) + (Eβ) + (FK) we obtain (UE) by Theorem 3.4. Using the elliptic Harnack inequality, one obtain in a standard way the oscillating inequality for harmonic functions and then for functions of the form u = GΩf (that solves the equation Ωu = f) in terms of f . Ω L k k∞ If now u = Pt f then u satisﬁes the equation d u = Ωu dt −L whence d u = G u . − Ω dt Knowing an upper bound for u, that follows from the upper bound for the heat ker- d nel, one obtains also an upper bound for dt u in terms of u. Applying the oscillation inequality one obtains the H¨oldercontinuity of u and, hence, of the heat kernel. Let us prove the on-diagonal lower bound

α/β pt (x, x) ct− . ≥

As in the proof of Theorem 2.2, (UE) and (Vα) imply that

1 pt (x, y) dµ (y) ≥ 2 ZB(x,r) 4.2. MATCHING UPPER AND LOWER BOUNDS 49 provided r Kt1/β. Choosing r = Kt1/β, we obtain ≥ 2 p2t (x, x) = pt (x, y) dµ (y) ZM 1 2 pt (x, y) dµ (y) ≥ µ (B (x, r)) ZB(x,r) c c = 0 . ≥ rα tα/β Then (NLE) follows from the upper estimate for

pt (x, x) pt (x, y) | − | when y close to x, which follows from the oscillation inequality. Assuming that the heat kernel exists, deﬁne the Green kernel g (x, y) by

∞ g (x, y) = pt (x, y) dt. Z0 If the Green kernel is ﬁnite then it is the integral kernel of the Green operator 1 G = − . If the heat kernel satisﬁes (UE) and (NLE) and α > β (a strongly transientL case), then it follows that

β α g (x, y) d (x, y) − . ((G)) ' n 2 n For example, in R we have g (x, y) = cn x y − , n > 2. | − | Corollary 3.7. (The transient case) Assume α > β > 1. If (Vα) is satisﬁed then (G) (UE) + (NLE) ⇔

In the proof one veriﬁes that (G) (H) + (Eβ). ⇒ 4.2 Matching upper and lower bounds

The purpose of this section is to improve both (UE) and (NLE) in order to obtain matching upper and lower bounds for the heat kernel. The reason why (UE) and (NLE) do not match, in particular, why (NLE) contains no information about lower bound of pt (x, y) for distant x, y is the lack of chaining properties of the distance function, that is an ability to connect any two points x, y M by a chain of balls of controllable radii so that the number of balls in this chain∈ is also under control. For example, the chain condition considered above is one of such properties. If (M, d) satisﬁes the chain condition then as we have already mentioned, (NLE) implies the full sun-Gaussian lower estimate by the chain argument and the semigroup property. 50 CHAPTER 4. TWO-SIDED BOUNDS OF THE HEAT KERNEL

Here we consider a setting with weaker chaining properties. For any ε > 0 introduce a modiﬁed distance dε (x, y) by

dε (x, y) = inf d (xi, xi 1) ((8)) xi is ε-chain − { } i=1 X N where ε-chain is a sequence xi of points in M such that { }i=0

x0 = x, xN = y, and d(xi, xi 1) < ε for all i = 1, 2, ..., N. −

Clearly, dε (x, y) is decreases as ε increases and dε (x, y) = d (x, y) if ε > d (x, y). As ε 0, dε (x, y) increases and can go to or even become equal to . It is easy to ↓ ∞ ∞ see that dε (x, y) satisﬁes all properties of a distance function except for ﬁniteness, so that it is a distance function with possible value + . ∞ It is easy to show that

dε (x, y) εNε (x, y) , ' where Nε (x, y) is the smallest number of balls in a chain of balls of radii ε connecting x and y:

xi xN=y

x0=x

Figure 4.1: Chain of balls connecting x and y

Nε can be regarded as the graph distance on a graph approximation of M by an ε-net. If d is geodesic then the points xi of an ε-chain can be chosen on the shortest { } geodesic, whence dε (x, y) = d (x, y). If the distance function d satisﬁes the chain d(x,y) condition then one can choose in (8) an ε-chain so that d (xi, xi+1) C , whence ≤ N dε (x, y) Cd (x, y). In general, dε (x, y) may go to as ε 0, and the rate of ≤ ∞ → growth of dε (x, y) as ε 0 can be regarded as a quantitative description of the chaining properties of d.→ We need the following hypothesis 4.2. MATCHING UPPER AND LOWER BOUNDS 51

Cβ Chaining property: for all x, y M, ∈ β 1 ε − dε (x, y) 0 as ε 0, → → or equivalently, β ε Nε (x, y) 0 as ε 0. → → β 1 For x = y we have ε − dε (x, y) as ε which implies under (Cβ) that there is ε 6= ε (t, x, y) that satisﬁes the→ ∞ identity→ ∞

β 1 ε − dε (x, y) = t ((9))

(always take the maximal possible value of ε). If x = y then set ε (t, x, x) = . ∞ Theorem 3.8. If (Vα) + (Eβ) + (H) and (Cβ) are satisﬁed then

1 β β 1 C dε (x, y) − pt (x, y) exp c ((10)) tα/β − t ! C exp ( cNε (x, y)) , ((11)) tα/β − where ε = ε (t, x, y).

Since dε (x, y) d (x, y), the upper bound in (10) is an improvement of (UE); similarly the lower≥ bound in (10) is an improvement of (NLE). The proof of the upper bound in (10) follows the same line as the proof of (UE) with careful tracing all places where the distance d (x, y) is used and making sure that it can be replaced by dε (x, y). The proof of the lower bound in (11) uses (NLE) and the semigroup identity along the chain with Nε balls connecting x and y. Finally, observe that (10) and (11) are equivalent, that is

1 β β 1 dε (x, y) − Nε , ' t β 1 which follows by substituting here Nε dε/ε and t = ε − dε (x, y) . ' Example. A good example to illustrate Theorem 3.8 is the class of post critically finite (p.c.f.) fractals. For connected p.c.f. fractals with regular harmonic structure the heat kernel estimate (11) was proved by Hambly and Kumagai ’99. In this setting d (x, y) is the resistance metric of the fractal M and µ is the Hausdorff measure of M of dimension α := dimH M. Hambly and Kumagai proved that (Vα) and (Eβ) are satisfied with β = α + 1. The condition (Cβ) follows from their estimate

d (x, y) β/2 Nε (x, y) C , ≤ ε because β β/2 β/2 ε Nε (x, y) Cd (x, y) ε 0 as ε 0. ≤ → → 52 CHAPTER 4. TWO-SIDED BOUNDS OF THE HEAT KERNEL

The Harnack inequality (H) on p.c.f. fractals was proved by Kigami ’01. Hence, Theorem 2.8 applies and gives the estimates (10)-(11). The estimate (11) means that the diﬀusion process goes from x to y in time t in the following way. The process ﬁrstly “computes” the value ε (t, x, y), secondly “detects” a shortest chain of ε-balls connecting x and y, and then goes along that chain.

Figure 4.2: Two shortest chains of ε-ball for two distinct values of ε provide diﬀerent routes for the diﬀusion from x to y for two distinct values of t.

This phenomenon was ﬁrst observed by Hambly and Kumagai on p.c.f. fractals, but it seems to be generic. Hence, to obtain matching upper and lower bounds, one needs in addition to the usual hypotheses also the following information, en- coded in the function Nε (x, y): the graph distance between x and y on any ε-net approximation of M. Example of computation of ε. Assume that the following bound is known for all x, y M and ε > 0 ∈ d (x, y) γ Nε (x, y) C , ≤ ε where 0 < γ < β, so that (Cβ) is satisﬁed (since Nε d (x, y) /ε, one must have β ≥ γ 1). Since by (9) we have ε Nε t, it follows that ≥ ' d (x, y) γ εβ ct, ε ≥ whence 1 t β γ ε c − . ≥ d (x, y)γ Consequently, we obtain

γ γ γ β β γ β γ − γ γ γ d (x, y) − d (x, y) Nε (x, y) Cd (x, y) ε− Cd (x, y) = ≤ ≤ t t ! 4.3. FURTHER RESULTS 53

and γ β β γ c d (x, y) − pt (x, y) α/β exp . ≥ t − ct !    Similarly, the lower estimate of Nε d (x, y) γ Nε (x, y) c ≥ ε implies an upper bound for the heat kernel

γ β β γ C d (x, y) − pt (x, y) α/β exp . ≤ t − Ct !   

Remark. Assume that (Vα) holds and all balls in M of radius r0 are connected, ≥ for some r0 > 0. We claim that (Cβ) holds with any β > α. The α-regularity of measure µ implies by the classical ball covering argument, that any ball Br of radius r α r can be covered by at most C balls of radii ε (0, r). Consequently, if Br ε ∈ is connected then any two points x, y Br can be connected by a chain of ε-balls r α ∈ containing at most C ε balls, so that r α Nε (x, y) C . ≤ ε Since any two points x, y M are contained in a connected ball Br (say, with ∈ r = r0 + d (x, y)), we obtain

β β α α ε Nε (x, y) Cε − r 0 ≤ → as ε 0, which was claimed. →

4.3 Further results

We discuss here some consequences and extensions of the above results.

Corollary 3.9. If (M, d) satisﬁes the chain condition then (Vα) + (Eβ) + (H) is equivalent to the two-sided estimate

1 β β 1 C d (x, y) − pt (x, y) exp c . ((12)) tα/β − t !

Proof. The implication

(Vα) + (Eβ) + (H) (12) ⇒ 54 CHAPTER 4. TWO-SIDED BOUNDS OF THE HEAT KERNEL

holds by Theorem 3.8 because dε d. For the opposite implication observe that '

(12) (Vα) ⇒ by Theorem 2.2 and

(12) (UE) + (NLE) (Eβ) + (H) ⇒ ⇒ by Theorem 3.6.

Conjecture. The condition (Eβ) above can be replaced by

β λmin (B (x, r)) r− . ((λβ)) '

In fact, (Eβ) in all statements can be replaced by the resistance condition:

β α res(Br,B2r) r − ((resβ)) ' where Br = B (x, r). In the strongly recurrent case α < β it alone implies the elliptic Harnack inequality (H) so that heat kernel two sided estimates are equivalent to (Vα)+(resβ) as was proved by Barlow, Coulhon, Kumagai ’05 (in a setting of graphs) and was discussed in M. Barlow’s lectures. An interesting (and obviously hard) question is characterization of the elliptic Harnack inequality (H) in more geometric terms - so far nothing is known, not even a conjecture. One can consider also a parabolic Harnack inequality (PHI), which uses caloric functions instead of harmonic functions. Then in a general setting and assuming the volume doubling condition (VD) (instead of (Vα)), the following holds:

(PHI) (UE) + (NLE) ⇔ (AG, Barlow, Kumagai in preparation). On the other hand, (PHI) is equivalent to

Poincar´einequality + cutoﬀ Sobolev inequality

(Barlow, Bass, Kumagai ’05).

Conjecture. The cutoﬀ Sobolev inequality here can be replaced by (λβ) and/or (resβ) .